A307872 Sum of the smallest parts in the partitions of n into 3 parts.
0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658, 707, 763, 812, 868, 932, 988, 1052, 1124, 1188, 1260, 1341, 1413, 1494, 1584, 1665
Offset: 1
Examples
Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
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n | 3 4 5 6 7 8 9 10 ...
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a(n) | 1 1 2 4 5 7 11 13 ...
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Links
- Michael De Vlieger, Table of n, a(n) for n = 1..3000
- Index entries for sequences related to partitions
Crossrefs
Cf. A069905.
Programs
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Mathematica
Table[Sum[Sum[k, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] Table[Total[IntegerPartitions[n,{3}][[;;,-1]]],{n,100}] (* Harvey P. Dale, Jan 14 2024 *) -
PARI
a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, k)); \\ Michel Marcus, May 02 2019
Formula
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} k.
Conjectures from Colin Barker, May 02 2019: (Start)
G.f.: x^3 / ((1 - x)^4*(1 + x)*(1 + x + x^2)^2).
a(n) = a(n-1) + a(n-2) + a(n-3) - 2*a(n-4) - 2*a(n-5) + a(n-6) + a(n-7) + a(n-8) - a(n-9) for n > 9.
(End)
a(n) = ((-1)^n*(-1+(-1)^r)+2*r*((-1)^(n+r)+(1+r)*(1+2*n-4*r)))/16, where r = floor(n/3). - Wesley Ivan Hurt, Oct 24 2021